Average Error: 15.4 → 0.0
Time: 1.2s
Precision: binary64
\[x + \left(1 - x\right) \cdot \left(1 - y\right)\]
↓
\[1 + \left(y \cdot x + 1 \cdot \left(-y\right)\right)\]

Error

Bits error versus x

Bits error versus y

Target

Original15.4
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1\right)\]

Derivation

  1. Initial program 15.4

    \[x + \left(1 - x\right) \cdot \left(1 - y\right)\]

  2. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left(x \cdot y + 1\right) - 1 \cdot y}\]

  3. Simplified0.0

    \[\leadsto \color{blue}{1 + y \cdot \left(x - 1\right)}\]
  4. Using strategy rm

  5. Applied sub-neg0.0

    \[\leadsto 1 + y \cdot \color{blue}{\left(x + \left(-1\right)\right)}\]

  6. Applied distribute-lft-in0.0

    \[\leadsto 1 + \color{blue}{\left(y \cdot x + y \cdot \left(-1\right)\right)}\]

  7. Final simplification0.0

    \[\leadsto 1 + \left(y \cdot x + 1 \cdot \left(-y\right)\right)\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.Vectors:renderPlotVectors from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (- (* y x) (- y 1.0))

  (+ x (* (- 1.0 x) (- 1.0 y))))